Quadratic algorithm to compute the Dynkin type of a positive definite quasi-Cartan matrix

Cartan matrices and quasi-Cartan matrices play an important role in such areas as Lie theory, representation theory, and algebraic graph theory. It is known that each (connected) positive definite quasi-Cartan matrix A ∈ M n ( Z ) A\in \mathbb {M}_n(\mathbb {Z}) is Z \mathbb {Z} -equivalent with the Cartan matrix of a Dynkin diagram, called the Dynkin type of A A . We present a symbolic, graph-theoretic algorithm to compute the Dynkin type of A A , of the pessimistic arithmetic (word) complexity O ( n 2 ) \mathcal {O}(n^2) , significantly improving the existing algorithms. As an application we note that our algorithm can be used as a positive definiteness test for an arbitrary quasi-Cartan matrix, more efficient than standard tests. Moreover, we apply the algorithm to study a class of (symmetric and non-symmetric) quasi-Cartan matrices related to Nakayama algebras.